curve #3652

y² + xy = x³ + x² − 34444685391230787101989206997x + 1860441465752852048025411799379070728824585

a-invariants: [1, 1, 0, -34444685391230787101989206997, 1860441465752852048025411799379070728824585]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 640259658391727630453646333422455570202860388634738
discriminant (Δ): 1120194582430750542556107143855195968763695868138947016775882261812725119593927140521972
Faltings height: 15.4135
naive height: 208.7411
primes of bad reduction: 2, 3, 7, 13, 23, 73, 157, 193, 211, 229, 277, 307, 367, 373, 421, 457, 701, 773, 1049, 3109, 5381, 22397
regulator: 219485420742320857713.64552901521859819704159046521439964807484576815
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:27:17 UTC
last updated: 2026-09-29 19:27:17 UTC

Witness: 15 independent points log in to add more points →

(-76781306939349455370328421814791/659840790694999104, 1110543775663006431774936022406845081478430618425/535992533030491756533960192)
(-1174218216835538341643620996106/21998242472622025, 6144816327328464528471630757209817296093753613/3262736316975264771941125)
(-141045541341465820945863790968/6742298596473001, 887840570532414249138210864507479486797892761/553620257946591418084501)
(-2245977562098000940123373/19214949924, 5516547045703547265479672681155472989/2663537928565032)
(-476517116657577293025932/2291632641, 19608724668173730760275265345040047/109702746157311)
(182480823600648126/841, 52321100539808121056555705/24389)
(1465938174524165456245/9840769, 5778104002262919700926037543745/30870492353)
(4699889619028607022606/27363361, 143945466870643181129436318529675/143137741391)
(130366775420850308239119/503778025, 36247412233520798160998183910539182/11307297771125)
(162807514315863148801820/652444849, 49446516440289948701601896215426445/16665398778007)
(189883103846819741314013105/1230699578161, 638721991239420374690963225660297415490/1365299960324890409)
(296847449011458386133856620/48490986821209, 433719145574976840006896122315148237049355/337669265805869468323)
(329224857035772081857992736720/2553805351969, 188902835179087726689881316884730963764478115/4081141842183636047)
(1030588967292859335471214284192/30461957711727151009, 229249640634256358703146752280058122707497076147315/168126710695187957095764414223)
(28691445333350186047401279422467/182200791030223716, 45432110367023165141513838792085529573561557531/77772379592329174520147736)

Commentary

Specialization at t = -379/743, u = 11/5 of the Elkies-Klagsbrun family y^2 = x^3 + 2A(t,u)x^2 + B(t,u)x (arXiv:2003.00077, Sec. 9, eqs. (3)-(4)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:27:17 UTC · history